let , and (will be referred to as axiom), and a string is embedded in a line where each bit not in is set to 0.

can be written as :

is the bit of the line.

given a line , where , we know that this can be seen as the pyramid shape that shows when we run , this fact will come in handy later.

in the other hand, we know axiomatically that all bits outside is set to 0, as seen by the white space outside the "pyramid".

we can reformulate as follows:

this writing is only useful to construct if we know and .
we know that for a string of length that, and
thus we can recursively reconstruct given and the axioms and .

After defining , answering our question becomes a simple task of recursively calling on the axiom .

The figure below is the result we reached, containing the first couple of lines of .

This result opens more questions than it answers, visually we can see that the newly generated section is structurally similar to the rest of .

On the other hand we can see that the closer we get to the axiom the more structured and predictable the strings become, visually this looks like a repeating pattern that becomes am infinite string of ones on the left and an infinite string of zeros on the right.

This section is entirely unsubstantiated, these are just some ideas to think about and may lead to interesting results:

• Is this new section reveals the repeating pattern of ?
• Is this the only possible pre axiom section possible?
• can we say anything about given ?

I thank Nathan George for his incredible obsidian plugin, it made life way easier, buy him a coffee!!